A Constructive Approach to Estimating Pure Characteristics Demand Models with Pricing Che-Lin Su The University of Chicago Booth School of Business Jong-Shi Pang University of Southern California Yu-Ching Lee speaker University of Illinois at Urbana-Champaign National Tsing Hua University (After Aug 2015) ICIAM 8, Aug 10 2015 , Beijing 

Development in theory of consumers discrete choice 1974: Discrete choice model proposed by McFadden: Products are viewed as set of characteristics. 1995: Random coefficients logit demand model proposed by Berry, Levinshon, and Pakes: Containing the taste for the characteristics and the taste for the product in consumer’s utility. The market share for a product is always positive, i.e., there are always benefits in introducing a new product. 2007: Pure characteristics demand model proposed by Berry and Pakes: The logit error term is removed from the utility function. Containing only the taste for the characteristics but not for the products. Implicitly imposing bound in introducing substitutive products. 

Traditional Way for Estimating PCM contraction mapping: computes an accurate solution only in specific cases element-by-element inverse: may not converge homotopy method: can be very slow We aim to develop a method of estimating that eliminates these restrictions. 

Contribution of our work 1. Formulate the pure characteristics demand model (PCM) estimation problem as a mathematical programming model. Form a Quadratic Program with Nonlinear Complementarity Constraints Propose a procedure to solve the estimation problem using existing solvers for the complementarity problem Obtain the Generalized Method of Moments (GMM) estimators 2. Resolve the computational burden in equating the true market share with the nonsmooth function of predicted market share. 3. Extend the market level taken into account in estimating the PCM. Not only fit into the observed quantities in the market but also the competitive environment defined as a Nash-Bertrand game 

Framework F firms, T markets, J products in each market, N sample consumers in each market, K characteristics for each product 1. Utility function of the Pure Characteristics Model Utility for consumer i buying product j in market t uijt = xjt T βi − αipjt + ξjt, where xjt ∈ RK : observed product characteristics, pjt : price of product j in market t, βi and αi: consumer specific coefficients, ξjt: characteristics that consumers observe but the model developer does not. Consumer i chooses to buy project j only if it provides the maximum and positive utility. j = 0 is called the outside option (e.g., buys nothing) and ui0t := 0. There is only one unobserved characteristics: ξjt. 

Framework (continued) 2. Nash-Bertrand game F + 1 players F firms which produce non-substitutive products aim at maximizing profit A virtual league of consumers over all the markets aims at maximizing aggregated utilities 3. Observed quantities quantity of product sold in each market (qjt), population in each market (Mt), market share for each product (sjt), and product price (pjt) 

Framework (continued) 4. Structural parameters βik = ¯ βk + σβkηik and αi = exp(¯ αwi) 5. Structural marginal cost mcjt = yT jt φ + ωjt 6. Consumer’s choice probability πijt ∈ [0, 1] 

Consumer’s choice We can define a probability tuple {πijt}J j=0 to represent the probability for consumer i to buy product j in market t. maximize {πijt}J j=0 J j=1 πijt(xjt βi − αipjt + ξjt) subject to J j=0 πijt = 1 and πijt ≥ 0, j = 0, . . . , J. 

Consumer’s choice It is easily verified that this purchasing mechanism can be written by 0 ≤ πijt ⊥ γit − [xT jt βi − αipjt + ξjt] ≥ 0, 0 ≤ γit ⊥ πi0t = 1 − J j=1 πijt ≥ 0, where γit captures consumer i’s maximum utility over all products in market t at the price tuple p and γit = max 0, max 1≤ ≤J xT t βi − αip t + ξ t . 

Firm’s pricing problem The original firm f’s pricing problem maximize pf ,πf ,γ T t=1 Mt N N i=1 j∈Jf πijt [ pjt − mcjt ] subject to for all t = 1, · · · , T; i = 1, · · · , N; and j ∈ Jf : 0 ≤ πf ijt ⊥ γit − xjt βi − αi pjt + jt ≥ 0 0 ≤ γit ⊥ 1 − J j =1 πf ij t ≥ 0 and pjt ≥ mcjt, ∀ j ∈ Jf ; ∀ t = 1, · · · T, where Mt : populations in market t, Jf : set of products produced by firm f, and mcjt : marginal cost of product j in market t. 

Firm’s pricing problem—Equivalent Formulation can be written equivalently as: maximize pf T t=1 Mt N N i=1 j∈Jf πijt min rij0t, min 1≤ ≤J rij t(p t) − mcjt subject to mcjt ≤ pjt, ∀ j ∈ Jf ; ∀ t = 1, · · · T where rij t(p t): pseudo-price, can be interpreted as the price to which product j adjusted. At the pseudo-price, consumer i’s utility of buying product j is the same as buying product given price of . That is, xT jt βi − αirijlt(plt) + ξjt = xT t βi − αip t + ξ t and rij0t xT jt βi + ξjt αi . 

Bound on price Moreover, the following constraint is added to the model to avoid the firm setting the price arbitrarily high: N i=1 T t=1 j∈Jf xjt βi − αi pjt + ξjt ≥ δf , 

Market optimization problem Combining with the maximizing sum of utilities problem: maximize π ≥ 0 T t=1 N i=1    J j=1 πijt xjt βi − αi pjt + jt    subject to J j=1 πijt ≤ 1, ∀ i = 1, · · · , N; ∀ t = 1, · · · , T. 

Market optimization problem we form a Nash-Betrand game LCPNB(Mt, N, α, β, x, mc, ξ): 0 ≤ vijt ⊥ Mt N πijt − J =1 λij t ≥ 0 0 ≤ pjt ⊥ − N i=1 j ∈Jf λij jt + N i=1 αiµf ≥ 0 0 ≤ λij t ⊥ vijt + p t − ζij t ≥ 0, = 1 · · · , J. 0 ≤ µf ⊥ −δf + N i=1 T t=1 j∈Jf [ αi (rij0t − pjt − mcjt) ] ≥ 0. This game has multiple equilibriums defined by product price and choice probability (p, π) pairs. 

Observed Quantities Given the market population Mt, number of draws N, the consumers purchasing probability πijt and the observed quantity qjt of product j sold in market t, the following equation should hold: qjt = Mt N N i=1 πijt. If given the observed market share Sjt instead of qjt, the following equation should hold: Sjt = 1 N N i=1 πijt. 

Observed Quantities In the introduction we mentioned the difficulty of equating the estimated market share sjt to the observed one. It means to find estimators Θ = {αi, βik} such that sj(xt, pt, ξt(Θ); Θ) = Sjt, where sj(xt, pt, ξt(Θ); Θ) ≈ 1 N N i=1 1 xT jt βi − αipjt + ξjt ≥ max 1≤ ≤J {xT t βi − αip t + ξ t , 0} . This is hard because of the existence of the nonsmooth indicator function 1.1 1The approximation of sj (xt , pt , ξt ; Θ) in (??) is formulated by the Monte Carlo simulation taking N draws of βi and αi . 

Generalized Method of Moments (GMM) Estimation QPCCEsP (Mt, N, q, pobs, x, y, η, w, (¯ α)): minimize θ ∈ Υ; mc; ξ; ω; z 1 2 ξ ZξWξZξ ξ + 1 2 ω ZωWωZω ω subject to • for all t = 1, · · · , T, j = 1, · · · , J, and f = 1, · · · , F : Mt N N i=1 πijt = qjt; pjt = pobs jt − mcjt • for all t = 1, · · · , T; i = 1, · · · , N; and j = 1, · · · , J : complementarity constraints in LCPNB • 0 ≤ mcjt ≤ pobs jt • βik = ¯ βk + σβk ηik for k = 1, . . . , K, • αi = exp(¯ α wi) and • mcjt = yjt φ + ωjt. 

Regularization Fixing an ¯ α, the estimation problem becomes locally solvable by SNOPT. In validating the estimation model, we add ¯ α − ¯ αinc 2 + K k=1 ¯ βk − ¯ βinc k 2 + σβk − σinc βk 2 in the objective function to identify parameters that are closed to the incumbent values ¯ αinc k , ¯ βinc k and σinc βk . 

Data generation procedure and Results of estimation 1. Determine N, T, J, K, F, and Jf . Select ¯ β ≤ ¯ β ≤ ¯ β, σβ ≤ σβ ≤ σβ, φ ≤ φ ≤ φ, and ¯ α. Generate η ∼ Normal(0, 1), then compute βik = ¯ βk + σβk ηik. Generate w ∼ Normal(0, 1), then compute αi = exp(¯ αwi). Generate x ∼ Uniform(0, 1) and y ∼ Uniform(0, 1). 2. Generate instrumental matrix Zξ ∈ R(T×J)×H. Generate ξjt ∈ null(Zξ). Generate instrumental matrix Zω ∈ R(T×J)×H. Generate ωjt ∈ null(Zω). Calculate mcjt = yT jt φ + ωjt. 3. Solve the LCPNB. Obtain qjt = Mt N N i=1 πijt and pobs jt = ˆ pjt + mcjt. 

Data generation procedure and Results of estimation 4. Generate Wξ and Wω. Solve QPCCEsP with fixed ¯ α. Instance Description Instance Size Identified by SNOPT SNOPT Solving Statistics Comment on and Sol Name N J K T F H Pricing Prob. # of comp. # of equal. Obj Value Time (sec) Iterat. Status ME1 5 4 11 3 2 12 MP1 (zero) 389 595 8.97E-10 0.33 1056 optimal all , =0 ME2 5 4 11 3 2 12 MP2 (not in null) -5.46E-12 0.42 1284 optimal all , =0 ME3 10 4 4 5 2 20 MP3 (zero) 1272 1512 9.04E-10 2.17 2651 optimal all , =0 ME4 10 4 4 5 2 20 MP4 (not in null) 2.1598 1.93 2465 optimal All =0 ME5 10 4 4 5 2 20 MP5 (not in null) 9.23E-06 2.04 2559 optimal The largest is 0.000524, all is zero. (For those with objective value 0, the incumbent parameters are recovered.) 

Data generation procedure and Results of estimation 5. Line search on different values of ¯ α. Instance Description SNOPT Solving Statistics Parameters Estimation MSE Name Special Handling Obj Value Time(sec) Iteration Status E21 as ME4 but fix = 0.1 -3.64E-12 2.7 3329 optimal 0.00E+00 8.63E-33 0.00E+00 E22 as ME4 but fix = 0.5 -1.82E-12 1.86 2389 optimal 0.00E+00 3.87E-29 0.00E+00 E23 as ME4 but fix = 0.9 28.88 2.01 2559 optimal 5.96 0.21 0.00E+00 E24 as ME4 but fix = 1.1 28.88 2.03 2559 optimal 5.96 0.21 0.00E+00 E25 as ME4 but fix = 1.5 978.30 2.06 2615 optimal 176.91 2.61 0.00E+00 E26 as ME4 but fix = 2 55607.57 1.90 2395 optimal 176.91 2.61 0.00E+00 (Some ¯ α pushes the objective value down to 0.) 

Conclusion In this work, we have 1. Enriched the modeling of the estimating problem of the Pure Characteristics Demand Model 2. Established computational tractability via existing numerical algorithms for solving the Quadratic Program with Linear Complementarity Constraints (QPCC)